On the frequency of partial quotients of regular continued fractions

Transcription

1 On the frequency of partial quotients of regular continued fractions Ai-Hua Fan, Lingmin Liao, Ji-Hua Ma To cite this version: Ai-Hua Fan, Lingmin Liao, Ji-Hua Ma. On the frequency of partial quotients of regular continued fractions. Mathematical Proceedings, Cambridge University Press (CUP), 200, 48, pp < <hal > HAL Id: hal Submitted on 7 Jun 2009 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA Abstract. We consider sets of real numbers in [0, ) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by /2, are given by a modified variational principle.. Introduction Let Q c denote the set of irrational number. It is well-known that each x [0, ) Q c possesses a unique continued fraction expansion of the form x =, (.) a (x) + a 2 (x) + a 3 (x) +... where a k (x) N := {, 2, 3, } is the k-th partial quotient of x. This expansion is usually denoted by x = [a (x), a 2 (x), ]. For each j N, define the frequency of the digit j in the continued fraction expansion of x by whenever the limit exists, where τ j (x) := lim n τ j (x, n), n τ j (x, n) := Card{ k n : a k (x) = j}. This paper is concerned with sets of real numbers with prescribed digit frequencies in their continued fraction expansions. To be precise, let p = (p, p 2,...) be a probability vector with p j 0 for all j N and j= p j =, which will be called a frequency vector in the sequel. Our purpose is to determine the Hausdorff dimension of the set E p := {x [0, ) Q c : τ j (x) = p j j }. Let us first recall some notation. For any a, a 2,, a n N, we call I(a, a 2,, a n ) := {x [0, ) : a (x) = a, a 2 (x) = a 2,, a n (x) = a n } a rank-n basic interval. Let T : [0, ) [0, ) be the Gauss transformation defined by T(0) = 0, T(x) = /x (mod ) for x (0, ). For a given frequency vector p = (p, p 2,... ), we denote by N( p) the set of T- invariant ergodic probability measures µ such that log x dµ < and µ(i(j)) = p j for all j. (.2)

3 2 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA Let h µ stand for the measure-theoretical entropy of µ, and dim H A for the Hausdorff dimension of a set A. The main result of this paper can be stated as follows. Theorem.. For any frequency vector p, one has { } dim H (E p ) = max 2, h µ µ N( p) 2, log x dµ where the is set to be zero if N( p) =. By virtue of log T (x) = 2 log x, we see that 2 log x dµ is the Liapunov exponent of the measure µ. Therefore, the in the above is a variational formula which relates the Hausdorff dimension to the entropy and Liapunov exponent of measures. Theorem. provides a complete solution to the long standing problem that requests an exact formula for dim H (E p ). Let us recall some partial results in the literature. In 966, under the condition that j= p j log j <, Kinney and Pitcher [9] showed that dim H (E p ) j= p j log p j 2, log x dµ p where µ p is the Bernoulli measure on [0, ] defined by µ p (I(a, a 2,, a n )) = n p aj. The above lower bound is just the Hausdorff dimension of the Bernoulli measure µ p. However, by a result of Kifer, Peres and Weiss in 200, this is not an optimal lower bound. Indeed, it is shown in [8] that, for any Bernoulli measure µ p, j= dim H µ p 0 7 This surprising fact indicates that the collection of Bernoulli measures are insufficient for providing the correct lower bound for dim H (E p ). In 975, under the same condition j= p j log j <, Billingsley and Henningsen [2] obtained an improved lower bound dim H (E p ) µ N( p) h µ 2 log x dµ. Moreover, they proved that, for any fixed N N, this lower bound is the exact Hausdorff dimension of the set {x E p : a n (x) N for all n } provided that p j = 0 for all j > N. It is therefore quite natural to guess that this lower bound is the right value for dim H E p in general. However, as will be shown in Theorem., this is not the case. Actually, the lower bound due to Billingsley and Henningsen is only a half of the correct lower bound. The other half of the lower bound, namely, dim H (E p ) /2, can be proved basing on Lemma 2.4 in []. However, we will give a direct proof in this paper. The upper bound estimate is more difficult. In its proof, we will use techniques from [] and [2] to estimate the lengths of basic intervals. Not incidentally, an entropy-involved combinatorial lemma borrowed from [5] (see Lemma 2.7) will play an important role.

4 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS3 The paper is organized as follows. In Section 2, we give some preliminaries on the basic intervals and on the entropy of finite words. In Section 3, we establish the upper bound in Theorem.. In Section 4, we prove that dim H (E p ) /2 and show that we can drop the condition j= p j log j < in Billingsley and Henningsen s theorem and then obtain the lower bound in Theorem.. The last section serves as a remark. 2. Preliminary Let x = [a (x), a 2 (x), ] [0, ) Q c. The n-th convergent in the continued fraction expansion of x is defined by p n := p n(a (x),, a n (x)) q n q n (a (x),, a n (x)) =. a (x) + a 2 (x) a n (x) For ease of notation, we shall drop the argument x in what follows. It is known (see [6] p.9) that p n, q n can be obtained by the recursive relations: p =, p 0 = 0, p n = a n p n + p n 2 (n 2), q = 0, q 0 =, q n = a n q n + q n 2 (n 2). By the above recursion relations, we have the following results. Lemma 2. ([6]). Let q n = q n (a,, a n ) and p n = p n (a,, a n ), we have (i) p n q n p n q n = ( ) n ; n (ii) q n 2 n 2, a k q n n (a k + ). Lemma 2.2 ([4]). For any a, a 2,, a n, b N, b + 2 q n+(a,, a j, b, a j+,, a n ) q n (a,, a j, a j+,, a n ) Recall that for any a, a 2,, a n N, the set b + ( j < n). I(a, a 2,, a n ) = {x [0, ) : a (x) = a, a 2 (x) = a 2,, a n (x) = a n } is a rank-n basic interval. We write I for the length of an interval I. Lemma 2.3 ([0] p.8). The basic interval I(a, a 2,, a n ) is an interval with endpoints p n /q n and (p n + p n )/(q n + q n ). Consequently, one has I(a,, a n ) = q n (q n + q n ), (2.) and 2q 2 n I(a,, a n ). (2.2) Lemma 2.4. We have 8 I(x I(x,, j,, x n ) (j + ) 2,, ĵ,, x n ), where the notation ĵ means deleting the digit j. q 2 n

5 4 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA Proof. By Lemma 2.2 and (2.2), we have I(x,, j,, x n ) qn 2 (x,, j,, x n ) ( ) 2 2 qn 2 j + (x,, ĵ,, x n ) 8 I(x (j + ) 2,, ĵ,, x n ). We will simply denote by I n (x) the rank n basic interval containing x. Suppose that a n := a n (x) 2 and consider I n(x) = I(a,, a n, a n ) and I n(x) = I(a,, a n, a n + ) which are two rank n basic intervals adjacent to I n (x). By the recursive equation of q n and (2.), one has the following lemma. Lemma 2.5. Suppose that a n := a n (x) 2. Then the lengths of the adjacent intervals I n (x) and I n (x) are bounded by 3 I n (x) from below and by 3 I n (x) from above. For any x [0, ) \ Q and any word i i k N k, (k ), denote by τ i i k (x, n) the number of j, j n, for which a j (x) a j+k (x) = i i k. For N N, define Σ N := {,..., N}. We shall use the following estimate in [2]. Lemma 2.6 ([2]). Let N and n. For any x = [x, x 2, ] [0, ] Q c with x j Σ N for j n. Then for any k, we have log I n (x) 2 τ i i k (x, n)log p k(i,, i k ) q k (i,, i k ) n 2 k. (2.3) i i k Σ k N Now we turn to the key combinatorial lemma which will be used in the upper bound estimation. Let φ : [0, ] R denote the function φ(0) = 0, and φ(t) = t log t for 0 < t. For every word ω Σ n N of length n and every word u Σk N of length k, denote by p(u ω) the frequency of appearances of u in ω, i.e., p(u ω) = τ u(ω) n k +, where τ u (ω) denote the number of j, j n k +, for which Define ω j ω j+k = u. H k (ω) := We have the following counting lemma. u Σ k N φ(p(u ω)). Lemma 2.7 ([5]). For any h > 0, ǫ > 0, k N, and for any n N large enough, we have Card{ω Σ n N : H k (ω) kh} exp(n(h + ǫ)).

6 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS5 3. Upper bound 3.. Some Lemmas. Let (p(i,, i k )) i i k Nk be a probability vector indexed by N k. As usual, we denote by q k (a,, a k ) the denominator of the k-th convergent of a real number with leading continued fraction digits a,..., a k. Lemma 3.. For each k N and each probability vector (p(i,, i k )) i i k N k, p(i,, i k )log p(i,, i k ) p(i,, i k )log I(i,, i k ). i i k N k i i k N k Proof. Applying Jesen s inequality to the concave function log, we have p(i,, i k )log I(i,, i k ) p(i,, i k ) log I(i,, i k ) = 0. i i k N k i i k N k Lemma 3.2. Let p = (p, p 2,... ) be a probability vector and q = (q, q 2,... ) a positive vector. Suppose j= p j log q j = and j= qs j < for some positive number s. Then n j= lim p j log p j n n j= p s. j log q j Proof. This is a consequence of the following inequality (see [3], p.27): for nonnegative numbers s j ( j m) such that m j= s j = and any real numbers t j ( j m), we have m m s j (t j log s j ) log( e tj ). (3.) j= Fix n. Let s j = p j for j n and s n+ = j=n+ p j. Let t j = s logq j for j n and t n+ = 0. Applying the above inequality (3.) with m = n +, we get n n n s p j log q j p j log p j ( p j )log( p j ) log( + qj). s j= j= j=n+ j= j=n+ Consequently, n j= p j log p j n j= p s + ( j=n+ p j)log( j=n+ p j) j log q j n j= p + log( + n j= qs j ) j log q j n j= p. j log q j Using the facts j= p j log q j = and j= qs j letting n. j= <, we finish the proof by Lemma 3.2 implies the following lemma. Recall that Σ k N = {,...,N}k. Lemma 3.3. Let k. If (p(i,, i k )) i i k Nk is a probability vector such that p(i,, i k )log q k (i,, i k ) =, i i k N k then we have lim i i k Σ p(i k,, i k )log p(i,, i k ) N 2 i i k Σ p(i k,, i k )log q k (i,, i k ) 2. N

7 6 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA Proof. By Lemma 2., for any k N and any s > /2, we have q k (i,, i k ) 2s (i i k ) 2s = ( j 2s ) k <. i,,i k i,,i k j= Thus we get the result by Lemma Proof of the upper bound. To prove the upper bound, we shall make use of multi-step Markov measures. Let k, by a (k )-step Markov measure, we mean a T-invariant probability measure P on [0, ) satisfying the Markov property P(I(a,, a n )) P(I(a,, a n )) = P(I(a n k,, a n )) P(I(a n k,, a n )) (3.2) for all n and a,, a n N (see [3], p.9). We may regard a Bernoulli measure as a 0-step Markov measure. For each N 2, we denote by P k N = Pk N ( p) the collection of (k )-step Markov measures satisfying the condition N P(I(j)) = p j for j N and P(I(N)) = p j. (3.3) These Markov measures are ported by the set of continued fractions for which the partial quotients are bounded from above by N. For each i i k {, 2,, N} k, write p(i,..., i k ) = P(I(i,..., i k )). Put k p(i,, i k )log p(i,, i k ) α N,k := P P 2 p(i k N,, i k )log(p k (i,, i k )/q k (i,, i k )). (3.4) The argument in [2] (pp.7-72) shows that the following limit α N := lim k α N,k, exists and coincides with each of the following three limits: and Let α N := lim α N := lim α := lim k P P k N k P P k N α N := lim α N = lim j= p(i,, i k )log p(i,, i k ) 2 p(i,, i k )log q k (i,, i k ), p(i,, i k )log p(i,, i k ) p(i,, i k )log I(i,, i k ), k P P k N h P 2 log x dp. α N = lim α N = lim α N. (3.5) To prove the upper bound, we need only to prove the following two propositions. Proposition 3.4. For any N N large enough, we have { } dim H (E p ) max 2, α N.

8 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS7 Proposition 3.5. We have { } α max 2, h µ µ N( p) 2. log x dµ Remark that we will finally establish the formula in Theorem., thus by Proposition 3.4 and Proposition 3.5 we have { } { } { } max 2, liminf α N = max 2, lim α N = max 2, h µ µ N( p) 2, log x dµ and if liminf α N /2, then the limit of α N exists and equals to µ N( p) h µ 2 log x dµ. Proof of Proposition 3.5. By virtue of (3.5), we set Denote α = lim D := lim lim k P P k N lim k P P k N p(i,, i k )log p(i,, i k ) 2 p(i,, i k )log q k (i,, i k ). 2 p(i,, i k )log q k (i,, i k ). If D =, then by Lemma 3.3, we have α /2. Now pose that D <, which is equivalent to lim lim log x dp <. k By (3.5), α = lim P P k N lim k P P k N h P 2 log x dp. Without loss of generality, we may pose that there is a sequence of Markov measures P N,k P k N converging to a measure µ N( p) in the weak -topology such that h PN,k 2 log x dp N,k = α. lim lim k Then by the upper semi-continuity of the entropy function and the weak convergence, we have α µ N( p) h µ 2 log x dµ. Proof of Proposition 3.4. For any fixed integer N which is large enough, and any ǫ > 0, we have E p H n (ǫ, N), where H n (ǫ, N) := l= n=l { x [0, ) \ Q : τ j (x, n) n } p j < ǫ, j N.

9 8 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA For any γ > max {/2, α N } and for any integer k N, we have the γ-hausdorff measure (see [4], for the definition of H γ ) ( ) H γ H n (ǫ, N) where = τ j (x,n) n n=l p j <ǫ, j N n(p j ǫ)<m j<n(p j+ǫ), j N I n (x) γ ( n l) x x n A I(x,, x n ) γ, A := {x x n Σ n N : τ j(x x n ) = m j, j N}. (We recall that τ j (x x n ) denotes the times of appearances of j in x x n.) Let ñ := N j= m j. By Lemma 2.4, we have the following estimate by deleting the digits j > N in the first n partial quotients x,..., x n of x I n (x): n ñ I(x,, x n ) γ 8 (j + ) 2γ I(x,, xñ) γ, x x n A where j=n+ x xñ Ã Ã := { x xñ ΣñN : τ j (x xñ) = m j, j N }. Since γ > /2, for N large enough j=n+ 8 <. (3.6) (j + ) 2γ By applying Lemma 2.6, and noticing that τ i i k (x, ñ) τ i i k (x xñ) + k, we have Thus where I(x,, xñ) = exp{log I(x,, xñ) } exp 2 (τ i i k (x xñ) + k)log p k(i,, i k ) q k (i,, i k ) ñ 2 k. x xñ Ã i i k Σ k N m i i k x xñ B B := I(x,, xñ) γ exp 2γ i i k Σ k N (m i i k + k)log p k + 8γ + 8ñγ q k 2 k, { } x xñ Ã : τ i i k (x xñ) = m i i k i i k Σ k N. Take h = k i i k Σ k N ( ) mi i φ k ñ k + (3.7)

10 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS9 in Lemma 2.7. We have for any δ > 0 and for ñ large enough exp 2γ (m i i k + k)log p k + 8γ + 8ñγ q k 2 k x xñ B i i k Σ k N exp (h + δ) + 2γ (m i i ñ k + k)log p k + 8γ + 8ñγ q k 2 k. i i k Σ k N Rewrite the right side of the above inequality as where L(γ, k, m i i k ) := h + 2γ exp {ñ(l(γ, k, m i i k ))}, i i k Σ k N m i i k + k ñ log p k q k + 8γ ñ + 8γ 2 k + δ. Since there are at most (ñ k + ) Nk possible words of i i k in ΣñN, we have I(x,, xñ) γ x xñ à (ñ k + ) Nk exp { ñ ( L(γ, k, m i i k ) m i i k Notice that by the definition of à and B, the possible values of m i i k are restricted to satisfy the condition that the frequency of digit j in x xñ is about p j. Then when ñ, for i,, i k Σ k N )} m i i k ñ k + p(i,, i k ), (3.8) and {p(i,, i k ) : i,, i k Σ k N } defines a probability measure P in Pk N. Now take δ > 0 small enough and k large enough such that and γ > 8γ < δ, (3.9) 2k k p(i,, i k )log p(i,, i k ) + 5δ 2 p(i,, i k )log(p k (i,, i k )/q k (i,, i k )). (3.0) The last inequality comes from the definition of α N, (3.4) and the assumption γ > α N. By (3.8), for sufficiently large ñ, we have 8γ ñ < δ and k i i k Σ k N i i k Σ k N ( ) mi i φ k ñ k k m i i k + k ñ log p k q k φ(p(i i k )) i i k Σ < δ, k N p(i i k )log p k q k < δ. i i k Σ k N. (3.)

11 0 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA By (3.7) and (3.), L(γ, k, m i i k ) < k i i k Σ k N φ(p(i i k )) + 2γ i i k Σ k N Thus (3.0) implies L(γ, k, m i i k ) < 0. Hence finally, by (3.6), we can obtain for any γ > max {/2, α N }, ( ) H γ H n (ǫ, N) <. n=k This implies that dim H (E p ) max {/2, α N } as desired. p(i i k )log p k q k + 5δ. 4. Lower bound In this section, we first prove dim H (E p ) /2. Then we examine what happens if the condition j= p j log j < in Billingsley and Henningsen s theorem is violated. We will see that if the condition is not satisfied, then dim H (E p ) = /2 and N(p) =. The following is the key lemma for proving that dim H (E p ) /2. Lemma 4.. For any given sequence of positive integers {c n } n tending to the infinity, there exists a point z = (z, z 2,...) E p such that z n c n for all n. Proof. For any n, we construct a probability vector (p (n), p(n) 2,..., p(n) such that p (n) k k,...) > 0 for all k c n and c n p(n) k =, and that for any k, lim n p(n) k = p k. (4.) Consider a product Bernoulli probability P ported by n= {,...,c n}. For each digit k, consider the random variables of x N N, X i (x) = {k} (x i ), (i ). By Kolmogorov s strong law of large numbers (see [2] p.388), we have for each digit k, ( n ) n lim {k} (x i ) E( {k} (x i )) = 0 P a.s., n n i= i= which implies lim n n n i= {k} (x i ) = lim n n n i= p (i) k = p k P a.s.. (4.2) That is to say, for P almost every point in the space n= {,..., c n}, the digit k has the frequency p k. Considering each point in N N as a continued fraction expansion of a number in [0, ], we complete the proof. Proof of dim H (E p ) /2. Take c n = n in Lemma 4., we find a point z E p, such that For a positive number b >, set z n = a n (z) n ( n ). (4.3) F z (b) := {x [0, ) : a k 2(x) (b k2, 2b k2 ]; a k (x) = a k (z) if k is nonsquare}.

12 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS It is clear that F z (b) E p for all b >. We define a measure µ on F z (b). For n 2 m < (n + ) 2, set µ(i m (x)) = n b k2. (4.4) Denote by B(x, r) the ball centered at x with radius r. We will show that for any θ > 0, there exists b >, such that for all x F z (b), lim inf r 0 log µ(b(x, r)) log r θ. (4.5) 2 In fact, for any positive number r, there exist integers m and n such that I m+ (x) < 3r I m (x) and n 2 m < (n + ) 2. (4.6) By the construction of F z (b), a n 2(x) > b n2 >. Let x = [x, x 2, ]. By Lemma 2.5, B(x, r) is covered by the union of three adjacent rank n 2 basic intervals, i.e., B(x, r) I(x, x 2,, x n 2 ) I(x, x 2,, x n 2) I(x, x 2,, x n 2 + ). By the definition of µ, the above three intervals admit the same measure. Hence by (4.6), we have log µ(b(x, r)) log r log 3µ(I(x, x 2,, x n 2)) log 3 I(x, x 2,, x m+ ). (4.7) However, on the one hand, by (4.4) log µ(i(x, x 2,, x n 2)) = log n b k2 = On the other hand, by (2.2) and Lemma 2., we have n k 2 log b. (4.8) m+ log I(x, x 2,, x m+ ) log log(x k + ). Let us estimate the second term of the sum. First we have m+ n+ 2 log(x k + ) 2 Since x k 2 2b k2 for all k, we deduce m+ log(x k 2 + ) m+ log(x k 2 + ) + 2 log(z k + ). n+ n+ log(2b k2 + ) log(3b k2 ) By (4.3), since z n n, for all n, we know m+ n+ = (n + )log 3 + k 2 log b. log(z k + ) (n+) 2 log(k + ).

13 2 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA Thus log I(x, x 2,, x m+ ) n+ (n+) 2 log 2 + 2(n + )log k 2 log b + 2 log(k + ). (4.9) Combining (4.8) and (4.9), for any θ > 0, take b > to be large enough, we have lim inf n log µ(i(x, x 2,, x n 2)) log I(x, x 2,, x m+ ) 2 θ, x F z(b). Hence by (4.7), we obtain (4.5). Since θ can be arbitrary small, by Billingsley Theorem ([]), we have dim H (E p ) 2. Thus Theorem. is already proved under the condition j= p j log j <. Now assume j= p j log j =. Then for any invariant and ergodic measure µ such that µ(i(j)) = p j for all j, we have log x dµ(x) µ(i(j))log j = p j log j =. (4.0) j= This implies D = (see the proof of Proposition 3.5 for the definition of D). Then by Proposition 3.4, we have dim H (E p ) 2. Since we have already proved dim H (E p ) 2, we get dim H (E p ) = if p j log j =. 2 This is in accordance with the formula of Theorem. under the convention that = 0, because (4.0) implies N( p) =. Finally, we remark that N( p) = if and only if j= p j log j =. We have seen the if part. For the other part, assume j= p j log j <. Then the Bernoulli measure µ such that µ(i(j)) = p j satisfies log x dµ(x) µ(i(j))log(j + ) = p j log(j + ) <. j= which implies that N( p). j= 5. A remark As suggested by the referee, we add a remark on a problematic argument appearing in the literature. To obtain an upper bound of the Hausdorff dimension of a set, one usually applies the Billingsley s theorem by constructing a finite measure P on the set such that U s P(U) (see [4], p.67). For the set E p, where p j = 0 for some j, the Markov measure P satisfying (3.3) does not match because the cylinders starting with j do not j= j=

14 ON THE FREQUENCY OF PARTIAL QUOTIENTS OF REGULAR CONTINUED FRACTIONS3 charge the measure and the above inequality is obviously not true. This appeared unnoticed for long (see the remarks of Kifer [7], p. 202). This problem did exist in the proof of Theorem 2 in [2]. Let us briefly indicate how to get around the problem in the proof of Theorem 2 of [2] when p j = 0 for some j s. The basic idea is similar to that of Cajar (see [3], p.67) and that of Kifer [7]. Recall that P is a (k )-step Markov measure ported on the set {x [0, ) Q c : a n (x) N for all n }. It is uniquely determined by its values on the k-cylinders, namely, p(i,...,i k ) = P([i,..., i k ]), (i i k ) {, 2,, N} k. Let 0 < ǫ < and P ǫ be the perturbed (k )-Markov measure determined by p ǫ (i,...,i k ) = ( ǫ)p([i,..., i k ]) + ǫ N k, (i i k ) {, 2,, N} k. Now, we can apply the Billingsley s theorem with P ǫ to find an upper bound, and then get the desired result by letting ǫ 0. In the present paper, we have intentionally avoided using the Billingsley s theorem. The proof of upper bound consists of Propositions 3.4 and 3.5. Proposition 3.5 concerns some calculations for which the zero frequency of some digits will not cause any trouble. In the proof of Proposition 3.4, we have used a covering argument depending on the estimate (2.3) instead of using the Billingsley s theorem. This enables us to get the upper bound of the Hausdorff dimension. Acknowledgments : The authors are grateful to the referee for a number of valuable comments. This work was partially ported by NSFC (A. H. Fan) and NSFC07764 (J. H. Ma). References. P. Billingsley. Ergodic theory and information. (John Wiley and Sons, Inc., New York-London- Sydney 965). 2. P. Billingsley and I. Henningsen. Hausdorff dimension of some continued-fraction sets. Z. Wahrscheinlichkeitstheorie verw. Geb. 3 (975) H. Cajar. Billingsley dimension in probability spaces. Lecture Notes in Mathematics, 892 (Springer-Verlag, Berlin-New York, 98). 4. K. J. Falconer. Fractal Geometry, Mathematical Foundations and Application. (John Wiley & Sons, Ltd., Chichester, 990). 5. E. Glasner and B. Weiss. On the interplay between measurable and topological dynamics. Handbook of dynamical systems. Vol. B Elsevier B. V., Amsterdam. (2006) A. Ya. Khintchine. Continued Fractions. (University of Chicago Press, Ill.-London, 964). 7. Y. Kiefer. Fractal dimensions and random transformations. Trans. A.M.S. 348 (996) Y. Kifer, Y. Peres and B. Weiss. A dimension gap for continued fractions with independent digits. Israel J. Math. 24(), (200) J. R. Kinney and T. S. Pitcher. The dimension of some sets defined in terms of f-expansions. Z. Wahrscheinlichkeitstheorie verw. Geb. 4 (966) M. Iosifescu and C. Kraaikamp. The Metrical Theory on Continued Fractions. Mathematics and its Applications, 547 (Kluwer Academic Publishers, Dordrecht, 2002).. L. M. Liao, J. H. Ma and B. W. Wang. Dimension of some non-normal continued fraction sets. Math. Proc. Cambridge Philos. Soc. 45() (2008) A. N. Shiryaev. Probability, Second Edition. GTM 95 (Springer-Verlag, New York, 996). 3. P. Walters. An introduction to ergodic theory. (Springer-Verlag, New York-Berlin, 200). 4. J. Wu. A remark on the growth of the denominators of convergents. Monatsh. Math. 47(3) (2006)

15 4 AI-HUA FAN, LING-MIN LIAO, AND JI-HUA MA Ai-Hua FAN: Department of Mathematics, Wuhan University, Wuhan, , P.R. China & CNRS UMR 640-LAMFA, Université de Picardie Amiens, France address: Ling-Min LIAO: Department of Mathematics, Wuhan University, Wuhan, , P.R. China & CNRS UMR 640-LAMFA, Université de Picardie Amiens, France address: Ji-Hua MA: Department of Mathematics, Wuhan University, Wuhan, , P.R. China address: